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( it.4'v, P 378.794 G43455 WP-15 Division of Agricultural Sciences UNIVERSITY OF CALIFORNIA • .1: • NONOPTIMALITY OF PRICE BANDS IN STABILIZATION POLICY COU.S.Ci1011 .-CONOIANCS 5001C Nnl.kED 0'00. • VA.OG•101AE.S011\ '41f,111. 01?„011.10',A. 0re-0. 010V Virlit011 C.:LS. 551.48 0.1s1k,lt, 1,4110-SO1N ?Nkl., .19S4 7CUN:DATIO71 0;7 AGRIZ:ULTURAL. 7.C.:CNC." -,:y71C3 ,• \ Richard E. Just and Andrew Schmitz IVAITE DEPARTMelvf OF 2994 232 MEMORIAL 0001( AGRICULTURAL CIASSR0041 ANDCOLLECTION op p1 ice APPLIED 'kit stuc. Eco,",10411CS 80E0RDWilvEl?slrY or Artivivesor4 551 411,74'Esorfi08 c-\\ I. •• _ _ _ -1-Eiip . California AgisieuTEuia Giannini Foundation of Agricultural Economics December 1976 i Working Paper No. 15 NONOPTIMALITY OF PRICE BANDS IN STABILIZATION POLICY by Richard E. Just and Andrew Schmitz CT' C:ANNIN I AGFirZULTIMAL. I:CONCMICS NONOPTIMALITY OF PRICE BANDS IN STABILIZATION POLICY . Richard E. Just and Andrew Schmitz Richard E. Just is an apsistant professor and Andrew Schmitz is an associate professor of agricultural and resource economics at the University of California, Berkeley NONOPTEMALITY OF PRICE BANDS IN STABILIZATION POLICY* A considerable amount of research has been devoted to studying the welfare consequences of stabilizing prices of basic internationally traded commodities. A widely held theoretical conclusion in the price stabiliza- tion literature is that complete price stabilization is preferred to no stabilization, e.g., Samuelson, Hueth and Schmitz, and Turnovsky. However, these results are usually derived using the assumption of zero storage costs. Also, partial stabilization is not generally considered as an alternative to complete price stabilization. For these reasons, the theoretical stabilization literature has had a limited impact on the policy-making process. For example, if one considers complete price stabilization in reality, very high storage levels are required to enforce the policy; thus, storage costs usually became prohibitive. This is because artificially high price stability leads to high quantity variability of stodk operations. As an exception to the above, Massell (1969) showed that gains could be made from stabilization when storage costs are positive by using a "price band!" type of stabilization policy. With the price band policy, the buffer stock authority sets upper and lower price limits; buffer stock transactions are then made such that price can vary freely (without intervention) between the two limits, but it cannot move outside of these limits. • Since Massell's work appeared, the price band concept has became very popular in the many proposals developed in response to the world food crisis ,of-the-1370!s. For example, at the International-Wheat-Council discussions 2. in 1975 and 1976 (see Sarris for a discussion of these), many of the large wheat trading nations such as Canada, Australia, EEC, India, Egypt, and Japan proposed world wheat stabilization policies which operate according to a price band mechanism. Many economists have also advocated the price band proposed in world food price stabilization problems (e.g., Hillman, Johnson, and Gray). In addition, many of the recent simulation studies on the effect of price stabilization focus on price band policies (e.g., Sarris, Cochrane and Danin, and Sharples and Walker). A major purpose of this paper is to show that price band policies are, in general, inferior mechanisms for achieving optimal social welfare through stabilization (using the same criterion of social welfare as has been used previously). This is done by developing an alternative to the price band policy which turns out to provide a global optimum with respect to the form of the buffer stock intervention policy. For this reason, it is found that the policy proposed herein is also preferred to buffer stock rules which operate with production triggers such as the one proposed by Tweeten et al. and the U. S. proposal presented at the International Wheat Council discussions as an alternative to the plans cited above. The buffer stock policy suggested in this paper is one which modifies the demand curve by both rotation and stabilization and possibly altering the curvature such'that buffer stock transactions make up the difference in modified and actual demand. It is found, however, that the optimum policy among this class of policies can be implemented by a simple rule such that buffer stock transactions are a constant multiple of the difference in actual price and "normal" price. Furthermore, a procedure for modifying the buffer-authority's declared normal .price _is_cle.termined. so that buffer stocks 3. will not be excessively accumulated or depleted over sustained periods of time. It is interesting to note that a policy with these features was evaluated in a simulation study by Cochrane and Danin and was found to be preferable to no stabilization and also to the popular price band policy. The present paper, however, shows that their result is true analytically (simulation leaves room for doubt) and, furthermore, derives analytically the Optimum policy among this general class of policies. The Free-Market Model Suppose, as did Massell, that industry demand and supply are linear and stochastic. Let demand be represented by (I) p = D(q) = a - bq + where p is price, q is quantity, both a and b are fixed, and ô is random at the time of decision making. Let supply (or short-run industry mar- ginal cost) be represented by (2) C(q) = (1) + f3q + e where f3 and (I) are fixed and E is random at the time of private decisions. Assuming ordinary supply-and-demand conditions, both b and a are positive. Suppose, also, that production is random so that actual production q differs from planned production (10 because of weather or some other stochastic influence. Where the difference in planned and actual production is represented by (3) =q.•••••• •• • ••••• ••••• •••••• 4.• the probability distribution of e = (6 E )1 will be characterized byl a 6 E(e) = 0, E(ee') = 0 a > 0. 0 Assuming competition, planned production in a free market is thus determined where expected price equals expected marginal cost, (4) E(p) = E[C(q)]. Use of (4) implies from (1) and (2) that q = a - (I) 0 n (5) where n E a + b. Hence, free-market quantity and price are evident from (3) and (1): a - (f) + nc (6) . (7) P - Y + n(6 - where y = aa + 14). Partial Stabilization by Buffer Stock Operations Initially, the buffer policy is assumed to have the effect of modifying the demand curve both by reducing instability and by altering demand elasticity as in the earlier Massell (1970) paper. That is, the demand curve in (1) is assumed to be altered by buffer authority intervention obtaining-the- modified demand given by-- - 5. p =a -b q+ 0 0 (8) where a and b are alternative parameters set by the authority. 0 0 Also, . suppose 60 = k6 for some fixed k which is also controlled by the buffer authority. Since the quantity demanded for consumption according to (1) is (p) = (9) a - p and the modified quantity demanded for both consumption and storage from (8) is q*(p) = (10) a - p +6 o o b. the net change in stocks required to enforce (8) is d S = cl*(13) - q (p). Since the demand now perceived by producers is (8) rather than (1), planned production with partial stabilization qt can be determined by analogy with (5): ao n* = ' 10 (12) . where v = 44) V b The quantity traded by producers and price are thus evi0. dent from (3) and (8): ao q* — + . 6. (14) p* = where\O = a a0 + b0 (I). ) v(60 — ' 10 e The quantity demanded by consumers in (9) at p*, however, is -6 + v[a + (1 - k)6 bod q (p*) = (15) by Hence, from (11), net change in stocks is s (16) q* - qd(p*) = p s (n - %)) + (1 - k)8 where _ y ao b - a bo (17) by The short-run variance of stock transactions is thus 2 a 2 -I- (1 - k) o' (18) b 2 Buffer Stock. Transactions Costs The buffer authority in this model will generally incur transactions losses (or gains) by purchasing excess supplies at prices different from those at which excess demands are satisfied. Purchases of buffer stocks are given by P E p* s (sales are represented by P < 0) and have expected value (19) E(P) = pt + k(k - 1)a + b (b - b)a (S 0 0 7. where 8 Ps (20) Pt - Consumer Effects To evaluate the individual sector benefits of partial price stabiliz ation, the change in consumer and producer surplus must again be calculated. Following Massell (1969), the gain in consumer surplus is2 (p p*) d(p) d(p*) 2 which from (6), (7), (14), and (15) has short-run expected value 2 (21) 2 2 - n )a ÷ (1 - k) as E(Gc) = pc + 2b where 1 2 (22) (a ) b0 ii Ya - 0) by Producer Effects 910 Producer gains from partial stabilization are q* p* q* _ pq f( + 13x + E)dx and have expected value (23) _ E G P = p - g - 8. where _ 11 = g a0 v 4)) v / ftao (24) _ (a 0 a -0 A a ,nI nJ 0)2 (a 2 / f3, n2(a0 -)2 2v 2 a v2(a - 0)2 n Storage Cost Although Massell (1969) had previously considered the effects of price stabilization with positive storage cost in the framework of this paper, his framework does not reflect the cost of carrying many successive years of high production into many successive years of low production versus the cost of carrying alternating high productions into immediately following years of low production. In a later work by Just, storage costs were assumed to d pend at least indirectly on the length of time in storage. In this paper, however, this dependence is assumed explicitly so that storage costs during each time period are given by the storage cost per unit p times the quantity held in stocks during that time period, (25) = p S. Social Optimization The approach in this section is to suppose a fixed stock goal which the buffer authority continually attempts to move. toward The possibility of determining optimal buffer stock size S will then also be discussed. fixed stock goal is met in expectations when (26) The 9. where S is the current level of stocks. In other words the expected change in stocks should just bring buffer stocks back to their desired size. p is fixed, and either a or b can be eliminated using (17). s 0 0 Hence, Solving for a obtains 0 b v ps - gbo - b) (27) a a(a - b0) n • which implies that (28) = (a - .15 + b 6 (29) abil q5+f3a Hence, using (20), (22), (24), (28), and (29), it becomes apparent that b P 2 S. (30) (a a - (15)P t - ps) (2ab 2 4. (32) Pg 2 ab2 us2 + 2a 2n b ps) b(a - s 2 Before maximizing benefits with this policy, two additional considerations are needed. First, some constraint must be imposed to ensure that a buffer stock policy is not imposed which may (with nonnegligible probability) require more stocks than are currently available. policy may not actually be in effect anyway. Otherwise, the advertised Second, the repercussions of 10.' this period's buffer policy on future periods must also be considered. Sup- pose, for political reasons, that the buffer authority is constrained so that its attempted stock size must at least satisfy some constant (1) times the variance a of buffer authority transactions with whatever policy is s selected. If the distributions of and ô are such that only finite and 6 are possible, then this constraint may simply require that stocks can never be depleted by chance and, thus, render the buffer stock policy inoperative. The maximization problem, ignoring repercussions in future periods, thus becomes* Max k,bep E(G) = E(G )+ E(G ) - E(P) - C s subject to the constraint imposed on the buffer authority S = as. The constraint can be imposed by substituting for C according to (25) and (26) s which implies that Cs = P(g - P ) = P as - P us. Finally, consider the extent to which future periods are affected by this period's policy. „. carried over. Future periods are affected by the amount of stocks From (16) and (26), it is apparent that expected stocks for the next period are E(S + s) = S + ps = §, and the variance is as. Assum- ing the policy in (26) will also be followed next period, the desired stock transaction p* for the next period (corresponding to p for this period) s will be (34) p* = (S s) = - (s - _ (n s + (1 - k)S 11. The. effects of changes in the distribution of s on next period's expected gains can be examined by an analogy with this period's gains E(G) if ps is replaced by p: and and CS are serially uncorrelated. The only changes in the next period's expected gains which depend on this period's policy (i.e., on ,the distribution of p*) relate to the terms p p and p . Using asters t' c' isks to denote the next period's values in (30), (31), and (32), using (34) and taking expectations implies a E(pt) = 4ab n s - b E(pV = 2 E(p*) = b a a2 as 2 2 f3 b a 2 s 2 since E(p*) = 0 in (34). Hence, the next period's expected gains are E(G*) = K S P (D a where K is a constant with respect to this period's policy controls. Note* that expected storage costs for the succeeding period are E(Cise) = Etp(S + s)] = p since E(p) = ; thus, only p = p cD as as enters E(G*). Since the distributions of stocks beyond the next period are not affected by this period's policy when (26) is imposed, an appropriate objective 12. is thus to maximize E(G) social discount rate. E(G*)/(1 r) subject to (33) where r is the Kuhn-Tucker conditions indicate optimum gains where 3E(G) + 1 3E(G*) ak 1+r Dk 0, k 3E(G) ( ak 1 3E(G*)) = o 1+r ak (35) 3E(G) ÷ 1 n (G*)> 0 31) 1 r 31) — o 1 DE(G*) b 3E(G) + 0 3b ( 1+r al) = O. Using (19), (21), (23), (30)-(32), and (18) obtains ka DE(G) _ ak (36) b 3E(G) = _ ab o 0 s 3E(G*) ' 3k P a P 3a s 3b Sb . p 2n 3E(G*). 31)0 (3 b 2n (;) )3a s 3a s = 3k 3a 3a ) s s ab' . o 0 2(1 - k)a6 2 b 2(b - bo)a, b 2 The implicit simple calculus conditions in (35) can be imposed for maximization so long as the indicated optimum controls are.positive. One finds from 3E(G) 3k 3E(G*) 1 - 0, 1+ r 3k (G) 3b 3E(G*) 1 = 0 1+ r ab using (36) that (37) k = K = Bb + 2pric1)(2 + r) (31) + (1 + Onb + 2pn(D(2 + where obviously E, 17.0 > o. b = E K b 0 o It can also be verified that the appropriate second-order conditions hold throughout the necessary range (k, 130 > 0). 13. The Desirable Degree of Price Stability Note O<k < 1 and 0 < 0 < b for reasonable parameter (all positive) values and that greater market intervention corre sponds to smaller K and b0. That is, from (18) a more sensitive stock rule and large as result when K and E are close to zero. 0 Also, greater market intervention leads to greater price stability since Var(p*) = k2a + k2b2 a from (14), whereas Var(p) = (5 2 a + b a in (7). Complete price stabilization is achieved by this policy when the effective demand curve is modif ied and rotated to the point of complete stability and infinite elasticity (1 c = 0, 130 = 0). K= 1 and b = b correspond to no interventi on. 0 On the other hand, Prom (37), it is thus evi- dent that higher storage costs lead to less intervention and in the limit (as p to no intervention. Even zero storage costs, however, do not imply complete stabilization is desirable becau se the value of stocks held out of current production must be discounted if they are held for consumption in the future. The optimal variance of price is given by -2 k (a + b2a ) 6 with K given by (37). Hence, both the degree of intervention and the opti mum level of price stability tends to be greater as storage costs fall, as the discount rate rises, and as the supply elast icity (or slope a) falls.3 Opti- • mal price stability increases with demand elasticity (or slope b) if demand variability ao is large relative to production variability a but decreases in demand elasticity if a is relatively larg 4 e. It is interesting to note, however, that increased free-market variabilit y due to larger supply-or-demand variation (larger ae2 a V ad) has no effec t on the optimal controls K. and 1. • b although the optimal variance of price shoul d increase proportionally to 0 the variability of free-market price. 14. Optimal Stock Size The optimal stock size or stock goal toward which the buffer authority should attempt to move can be found through the constraint imposed in maximization, namely, g = . The optimal stock variance is given by a* = (1 - k)2 2 + a 113 ). Hence, optimal stocks are 2 2 2 (1)(1 + r) n b (a + a /b 2) (38) (13b + (1 + r)flb + 2pri(1)(2 + It is evident from (38) that larger production and demand variation and smaller storage cost lead to larger buffer stocks (on average). However, a change in variation of production costs has no effect on optimal buffer stock size. This happens because production expenses are only incurred on the basis of planned production rather than actual production and is a result noted previously for the case where one uses the ex post surplus concept employed here (Hazell and Scandizzo; Just). It is also interesting to note that optimal stocks are never zero unless one of the parameters a, b, r, or P approaches infinity or as = ab = 0. The rule in (38) thus provides some information as far as the length of time which should be used in building up a buffer stock depending on the length of run used in defining supply-and-demand parameters b and a. For example, if one begins a production period with no buffer stocks in an agricultural --7- 15. situation where short-run supply is supposedly inelastic, then (38) implies no stock accumulation should be attempted, at least in expectations (0 -* co implies §* 0). On the other hand, as a longer period of adjustment is considered, supply becomes more elastic; and examination of the partial derivative of g* with respect to 13 indicates that higher supply elasticity (or slope 13) leads to larger optimal stocks. One also finds that higher demand elasticity leads to higher optimal stocks if demand variability ao is large relative to production variability Q; however, lower stock levels are implied by elastic demand when a is relatively large. Implementation of a Stabilization Policy Consider practical application of the buffer policy in (16). After de- termining ; 0 and K, imposition of the modified demand curve in (8) would still appear to require the very difficult measurement of 6 as well as q. Hence, it is desirable to find some simple rules for action which might be followed by a buffer stock authority. For example, a common suggestion in agriculture (including the U. S. proposal at the International Wheat Council discussion) is that a certain percentage of the excess of any given crop over a normal crop should be saved for years of shortage. suggested by Tweeten, et al. This approach is With a similar rule for depletion of reserves, the modified demand curve would be represented by P= - bq ÷ 6 c(q - pq) where c is a constant indicative of the fixed percentage rule and p mean or normal) crop. Specifically, change in stocks is is the 16. In terms of the general modified demand curve in (8), this implies a = a - cp , b =b 0 q 0 • c, 60 = a. Obviously, this simple rule does not suffice for net gains maximiz ation in general since it forces k = 1. Suppose a modified demand curve represented by p = a - bq + cS + (P P is considered. In this case the buffer authority either reduces stocks by some constant multiple c/b of the excess of current price over normal price (when p > p ) or increases stocks by the same multiple of the excess of normal price over current price (when 9 > p). That is, (39) Hence, this price-oriented buffer policy is almost as simple in practic e as the quantity-oriented policy above. In terms of the general modified demand curve in (8), this now implies a+c (40) a0 = b 1+ c ' 0 Here, however, the full controls of the previous section can be attained by setting c = (1 - i)/K with appropriate definition of p b = 0 K b and 6 since, in that case, = K 6 in (40) as indicated in (37). This simple single- instrument, price-oriented policy can thus be used to obtain optimal values for both controlled parameters k and b and optimal net gains from partial 0 stabilization. 17. Optirality in a Broader Sense Although a linear modified demand curve was selected arbitrarily in (8), it can be shown that the controls in (37) or (39) are optimal in a much In point of fact, if one considers in place of (8) the class broader sense. of all nonlinear modified demand curves, it still turns out. that (37) or (39) give optimal controls so long as (1) and (2) are linear. Suppose for the moment that the distribution for (I) E 0, 0 is discrete with probability density function f(-) and that sample space points can be associated in pairs such that the probabilities satisfy ) i = 1, .••, n. ' 1Pi2 f(4)1.2) = Ei(q)i) = 0 n, includes the entire sample space.) 11)1.2, i = 1, Let Ei repre— sent the expectation operator for subdistribution i which has probability density function f(4)ii) f M) = 0 elsewhere. fi°P1.2) = 1 - fi") f OPi Now suppose that one considers the optimal stock and intervention policy problem of this paper for each subdistribution 1. Ei(G) E (G*)/(1+0, i = 1, 1 One would first compute n, and then maximize each with respect to the controls, say, k and bi. The only difference among these n maximi— i zation problems is in the variances of (3 and C since the same deterministic components of supply and demand would continue to apply. does not involve a 6 or a this implies that optimal controls for all sub— distributions are the same, i.e., Ki = K2 a =b. Thus, • 0 But since (37) =1-` 131 = 2 = •.• 1 • Max E(G) k,b 0 1 E(G*) = 1+r E P(i) Max E(G) i i=1 k b. . r E.(G*) Max Ei(G) + 1 E.(G*1 1 r a. k.,b. where P(i) is the probability associated with the ith subdistribution, f(IP) f(4). 2.2)- In intuitive terms, the result of the above proof is as follows. Sup- pose each of the n subdistributions represent successively higher degrees of variability. For example, in Figure 1 let = = q2 represent sample points in the first subdistribution and let ID 21 = c13, 22 = q4 represent sample points in the second subdistribution (for simplicity, assume f(q1) = f(q2) = 1/3, f(q3) = f(q4) = 1/6 and that E 0). Suppose that de- mand is represented by D, and the optimal modified demand curve corresponding to (40) for the overall problem is represented by D*. At productions ql, q2, q3, and q4, the optimal storage rule leads to prices pt, pI, p, and p, respectively, as opposed to respective free-market prices 131, p2, p3, and p4. Thebufferstocktransactionswiththestoragerulewouldbeav -av a3, and -a respectively. 4' Now suppose the first subdistribution with sample points (11 and q2 is considered separately. Again, one finds the optimal corresponding prices are p* and p* with buffer stock transactions a and respectively, since 1 2 1 2' E1 = EBut 0. note that, in the two-sample-point case, the optimal p* and p* are 1 2 not necessarily dependent on an assumption such as in (8). For example, one could consider the price band approach proposed by Massell (1969) for the case with positive storage costs. Optimizing his price band (choosing a maximum price p and a minimum price 2) implies that price limits should be p =-1Praridil:= p! since, with a twc5:-.0oiii-t distiibution, his 4proach is equivalent to (8). 19. Price P3 PI, PI 4 '2 2 ' 4 134 3 Figure 1. qi Pq q2 4 Comparison of Alternative Modified Demand Curve Quantity/u.t. 20. 1 Finally, consider the second subdistribution with sample points q3 and q4. The results presented above imply that the same modified demand curve D* should again be imposed; hence, prices corresponding to q3 and q4 should be p* and p*, respectively, with buffer transactions a and -a 3 3 4' respectively. However, choosing an optimal price band in this case would imply p = p* and 2. = p, 4 again because the two approaches became equivalent 3 with a two-point distribution. To summarize the above arguments, one finds that the optimal price band changes with variability (of production in this case). But the modified curve D* reaches optimality for all levels of variability simultaneously and, hence, price band rules must, in general, be suboptimal with respect to the storage rule in (37) or (39). Furthermore, these same conclusions are reached in comparing D* to any other nonlinear modified demand curve since modifications, such as and b* in Figure 1, are also possibilities when one considers maximization within each subdistribution separately. Moreover, when alter- native subdistributions with, say, ql and q4 in one and q2 and q3 in the other are considered, then nonlinear curves which are not inverse symmetric about p such as D** also become possibilities and, therefore, these must also be dominated by_the storage rule in (37) or (39). The arguments in this section have been presented in a simple approach with discrete distributions to add intuition. But by. formalizing them with the rigor of measure theory, they become applicable to continuous and mixed distributions as well. Hence, the simple price-oriented storage rule in (39) is optimal in a fairly broad sense and definitely dominates the popular price band approach. 21. Conclusions The model in this paper points out an important problem which the literature on stabilization has often failed to consider; namely, research has too often focused on price stabilization rather than economic stabilization. Too often, only two polar alternatives have been compared, while •a whole range of possibly preferable intermediate policies have been ignored. When one properly introduces storage costs, it becomes apparent that the quantity variability of stock transactions caused by complete price stabilization, indeed, significantly reduces benefits because a much larger buffer stock is thus required. Although more price stabilization may al- ways be preferable on the part of producers and consumers (combined), the costs incurred by the buffer authority in additional price stabilization are increasing at an increasing rate. The buffer authority is better off with quantity stabilization of buffer stock transactions. There is thus a trade-off between price stabilization and quantity stabilization, and the question of economic stabilization which should properly be addressed involves finding an optimal trade-off between the two. A price band policy can only attain the optimal trade-off when the distribution of supply or demand has two sample points as in the graphical results presented by Massell and Hueth and Schmitz. However, in this paper, it is found that a simple price-oriented pol.icy rule varying stock transactions proportionally to price movements can simultaneously attain the optimal trade-off for all levels of variability. It must be noted that the results in this paper are derived assuming linearity in free-market supply and demand and that disturbances are additive. However, it would seem that the question of finding a policy which provides 22. an optimal trade-off between price variability and quantity variability over the whole range of possibilities (simultaneously) would again be the important question under nonlinearity. Hence, it seems that price bands would not be optimal (except in special cases) under nonlinearity just as with linearity. Other generalizations which are also needed in this work are to introduce serial dependence of disturbances in supply and demand and to consider the possibility of only partial adjustment of stock levels toward desired stock goals. By considering the complete (but stochastic) adjustment each period with no serial correlation in this paper, it was possible to reduce the general stochastic control problem to one concerned with only two time periods. It stands to reason that larger social gains may also be attained by generalizing the control, but solution of the problem would be considerably more complicated. This approach would also help to further answer the question about how and at what rate buffer stocks should be accumulated initially. 23. Footnotes *Giannini Foundation Paper No. 1 Although the covariance matrix is assumed diagonal for ease of exposition; the results can be very easily generalized without substantial changes as demonstrated by Just (1975). 2 As in earlier papers on this subject, the analysis in this paper is based on the concept of economic surplus. The limitations of the surplus approach have been clearly pointed out by Currie, Murphy, and Schmitz and should be kept firmly in mind. 3 It seems counter-intuitive that increasing the social discount rate should lead to greater price stability since one would normally expect storage for future use to be less valuable in that case. This result occurs, however, because, with a greater discount rate, social gains depend more completely on the stability attained in the immediate production period rather than on the stability attained in the future. 4 These assertions can be verified by determining the sign of partial de-2 rivatives of k (a „. +b 2 a ). 24. Reference Cochrane, W. W., and Y. Danin. Reserve Stock Grain Models: The World and the United States, 1975-85. Minnesota Agr. Exp. Sta. Tech. Bull. 305, 1976. Currie, Martin, John Murphy, and Andrew Schmitz. "The Concept of Economic Surplus and Its Use in Economic Analysis." Econ. J. 81 (1971): 741-99. Hazell, P. B. R., and P. L. Scandizzo. "Optimal Price Intervention Policies When ' - roduction is Risky," in Risk and Uncertainty in Agricultural Development, eds. J. Roumasset, J. Boussard, and I. J. Singh. Berkeley:University of California Press (forthcoming). Hillman, Jimmye, D. Gale Johnson, and Roger Gray. Food Reserve Policies for World Food Security: A Consultant Study on Alternative Approaches. Food and Agric. Organization of the U. N. Hueth, Darrell, and Andrew Schmitz. Final Goods: J. Edon. (ESC:CSP/75/2), Rome, 1975. "International Trade in Intermediate and Some Welfare Implications of Destabilized Prices." Quart. 86 (1972):351-65. Just, Richard E. A Generalization of Some Issues in Stochastic Welfare Eco- nomics: Implications for Agricultural Price Stabilization. Oklahoma Agr. Exp. Sta. Res. Rept. P-712, Apr. 1975. Massell, Benton F. "Price Stabilization and Welfare." Quart. J. Edon. 83 (l969):285-97. "Some Welfare Implications of International Price Stabilization." J. Pol. Econ. Samuelson, Paul A. 78 (1970):404-17. "The Consumer Does Benefit from Feasible Price Stability." Quart. J. Edon. 86 (l972):476-98. Sarris, Alexander Hippocrates. "The Economics of International Grain Reserve Systems." Ph.D. thesis, MIT, 1976. 25. Sharpies, Jerry A., and Rodney L. Walker. Analysis of Wheat Loan Rates and Target Prices Using a Wheat Reserve Stocks Simulation Model. USDA ERS, Commodity Economics Division, Research Status Report Number 2, May, 1975. Turnovsky, S. J. "The Distribution of Welfare Gains from Price Stabilization: The Case of Multiplicative Disturbances." Int. Econ. Rev. 17(1976): 133-48. Tweeten, L., Dale Kalbfleisch, and Y. C. Lu. An Economic Analysis of Cdrxrd- over Policies for the United States Wheat Industry. Oklahoma Agr. Exp. Sta. Tech. Bull. T-132, Oct. 1971. United States, Public Affairs Office. "International Grain Reserves: Proposal." USA: Information. Affairs Office. 1975. U. S. U. S. Mission to OECD, Paris. Public No. 8 (1975); reprinted in Eastern Economist, Oct. 24,